From structural to temporal and spatial hierarchies in heteroclinic networks

We construct heteroclinic networks from generalized Lotka-Volterra equations of winnerless competition. Our heteroclinic orbits connect saddle-fixed points characterized by the temporary survival of single species. By an appropriate choice of rates in the predation matrix we implement a structural hierarchy in the attractor space that consists of iterated heteroclinic cycles of heteroclinic cycles. This induces a hierarchy in time scales that amounts to iterated oscillatory modulations of fast oscillations on slower time scales. We indicate how the temporal hierarchy translates to spatial scales by coupling these heteroclinic networks on a spatial grid. Our concrete realization then leads to rock-paper-scissors games, played on several scales simultaneously. Applications of this kind of heteroclinic dynamics are transient neuronal dynamics in the brain, possibly related to the binding problem, and chunking dynamics, or transient dynamics in ecological systems with games played in parallel on the scale of individual species as well as of populations and metapopulations.

M. Voit, H. Meyer-Ortmanns, arXiv:1806.11039 to be publ. in EJP ST 2018, edited by V. Nekorkin and V. Klinshov.